Properties

Label 1104.2.e.f.47.4
Level $1104$
Weight $2$
Character 1104.47
Analytic conductor $8.815$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1104,2,Mod(47,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1104.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.81548438315\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3814238552064.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 4x^{5} - 12x^{3} + 9x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 47.4
Root \(-0.696260 - 1.58595i\) of defining polynomial
Character \(\chi\) \(=\) 1104.47
Dual form 1104.2.e.f.47.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.696260 + 1.58595i) q^{3} +2.04950i q^{5} +2.04950i q^{7} +(-2.03044 - 2.20846i) q^{9} +O(q^{10})\) \(q+(-0.696260 + 1.58595i) q^{3} +2.04950i q^{5} +2.04950i q^{7} +(-2.03044 - 2.20846i) q^{9} -2.66837 q^{11} -5.30873 q^{13} +(-3.25040 - 1.42699i) q^{15} -6.71077i q^{17} +0.878038i q^{19} +(-3.25040 - 1.42699i) q^{21} -1.00000 q^{23} +0.799541 q^{25} +(4.91621 - 1.68251i) q^{27} -3.68294i q^{29} +3.90588i q^{31} +(1.85788 - 4.23188i) q^{33} -4.20046 q^{35} -0.668367 q^{37} +(3.69626 - 8.41936i) q^{39} -1.92686i q^{41} +9.63831i q^{43} +(4.52624 - 4.16140i) q^{45} +5.97710 q^{47} +2.79954 q^{49} +(10.6429 + 4.67244i) q^{51} +0.878038i q^{53} -5.46882i q^{55} +(-1.39252 - 0.611343i) q^{57} -14.4170 q^{59} -3.33673 q^{61} +(4.52624 - 4.16140i) q^{63} -10.8803i q^{65} -12.5659i q^{67} +(0.696260 - 1.58595i) q^{69} -8.43991 q^{71} -8.84593 q^{73} +(-0.556688 + 1.26803i) q^{75} -5.46882i q^{77} +3.22097i q^{79} +(-0.754597 + 8.96831i) q^{81} +9.28584 q^{83} +13.7537 q^{85} +(5.84093 + 2.56428i) q^{87} -13.7373i q^{89} -10.8803i q^{91} +(-6.19451 - 2.71951i) q^{93} -1.79954 q^{95} -11.2858 q^{97} +(5.41797 + 5.89298i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{9} - 4 q^{11} - 4 q^{13} + 10 q^{15} + 10 q^{21} - 8 q^{23} - 16 q^{25} + 12 q^{27} - 10 q^{33} - 56 q^{35} + 12 q^{37} + 24 q^{39} - 6 q^{45} - 8 q^{47} + 38 q^{51} - 16 q^{59} + 8 q^{61} - 6 q^{63} - 24 q^{71} - 20 q^{73} + 52 q^{75} + 2 q^{81} - 20 q^{83} - 24 q^{85} + 40 q^{87} + 2 q^{93} + 8 q^{95} + 4 q^{97} + 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(277\) \(415\) \(737\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.696260 + 1.58595i −0.401986 + 0.915646i
\(4\) 0 0
\(5\) 2.04950i 0.916565i 0.888807 + 0.458283i \(0.151535\pi\)
−0.888807 + 0.458283i \(0.848465\pi\)
\(6\) 0 0
\(7\) 2.04950i 0.774639i 0.921946 + 0.387319i \(0.126599\pi\)
−0.921946 + 0.387319i \(0.873401\pi\)
\(8\) 0 0
\(9\) −2.03044 2.20846i −0.676815 0.736154i
\(10\) 0 0
\(11\) −2.66837 −0.804543 −0.402271 0.915520i \(-0.631779\pi\)
−0.402271 + 0.915520i \(0.631779\pi\)
\(12\) 0 0
\(13\) −5.30873 −1.47238 −0.736189 0.676776i \(-0.763377\pi\)
−0.736189 + 0.676776i \(0.763377\pi\)
\(14\) 0 0
\(15\) −3.25040 1.42699i −0.839249 0.368446i
\(16\) 0 0
\(17\) 6.71077i 1.62760i −0.581144 0.813801i \(-0.697395\pi\)
0.581144 0.813801i \(-0.302605\pi\)
\(18\) 0 0
\(19\) 0.878038i 0.201436i 0.994915 + 0.100718i \(0.0321139\pi\)
−0.994915 + 0.100718i \(0.967886\pi\)
\(20\) 0 0
\(21\) −3.25040 1.42699i −0.709295 0.311394i
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0.799541 0.159908
\(26\) 0 0
\(27\) 4.91621 1.68251i 0.946126 0.323799i
\(28\) 0 0
\(29\) 3.68294i 0.683904i −0.939717 0.341952i \(-0.888912\pi\)
0.939717 0.341952i \(-0.111088\pi\)
\(30\) 0 0
\(31\) 3.90588i 0.701516i 0.936466 + 0.350758i \(0.114076\pi\)
−0.936466 + 0.350758i \(0.885924\pi\)
\(32\) 0 0
\(33\) 1.85788 4.23188i 0.323415 0.736676i
\(34\) 0 0
\(35\) −4.20046 −0.710007
\(36\) 0 0
\(37\) −0.668367 −0.109879 −0.0549394 0.998490i \(-0.517497\pi\)
−0.0549394 + 0.998490i \(0.517497\pi\)
\(38\) 0 0
\(39\) 3.69626 8.41936i 0.591875 1.34818i
\(40\) 0 0
\(41\) 1.92686i 0.300925i −0.988616 0.150462i \(-0.951924\pi\)
0.988616 0.150462i \(-0.0480763\pi\)
\(42\) 0 0
\(43\) 9.63831i 1.46983i 0.678160 + 0.734915i \(0.262778\pi\)
−0.678160 + 0.734915i \(0.737222\pi\)
\(44\) 0 0
\(45\) 4.52624 4.16140i 0.674733 0.620345i
\(46\) 0 0
\(47\) 5.97710 0.871850 0.435925 0.899983i \(-0.356421\pi\)
0.435925 + 0.899983i \(0.356421\pi\)
\(48\) 0 0
\(49\) 2.79954 0.399934
\(50\) 0 0
\(51\) 10.6429 + 4.67244i 1.49031 + 0.654273i
\(52\) 0 0
\(53\) 0.878038i 0.120608i 0.998180 + 0.0603039i \(0.0192070\pi\)
−0.998180 + 0.0603039i \(0.980793\pi\)
\(54\) 0 0
\(55\) 5.46882i 0.737416i
\(56\) 0 0
\(57\) −1.39252 0.611343i −0.184444 0.0809743i
\(58\) 0 0
\(59\) −14.4170 −1.87693 −0.938467 0.345368i \(-0.887754\pi\)
−0.938467 + 0.345368i \(0.887754\pi\)
\(60\) 0 0
\(61\) −3.33673 −0.427225 −0.213613 0.976918i \(-0.568523\pi\)
−0.213613 + 0.976918i \(0.568523\pi\)
\(62\) 0 0
\(63\) 4.52624 4.16140i 0.570253 0.524287i
\(64\) 0 0
\(65\) 10.8803i 1.34953i
\(66\) 0 0
\(67\) 12.5659i 1.53516i −0.640951 0.767582i \(-0.721460\pi\)
0.640951 0.767582i \(-0.278540\pi\)
\(68\) 0 0
\(69\) 0.696260 1.58595i 0.0838199 0.190925i
\(70\) 0 0
\(71\) −8.43991 −1.00163 −0.500816 0.865554i \(-0.666967\pi\)
−0.500816 + 0.865554i \(0.666967\pi\)
\(72\) 0 0
\(73\) −8.84593 −1.03534 −0.517669 0.855581i \(-0.673200\pi\)
−0.517669 + 0.855581i \(0.673200\pi\)
\(74\) 0 0
\(75\) −0.556688 + 1.26803i −0.0642809 + 0.146419i
\(76\) 0 0
\(77\) 5.46882i 0.623230i
\(78\) 0 0
\(79\) 3.22097i 0.362387i 0.983447 + 0.181194i \(0.0579960\pi\)
−0.983447 + 0.181194i \(0.942004\pi\)
\(80\) 0 0
\(81\) −0.754597 + 8.96831i −0.0838441 + 0.996479i
\(82\) 0 0
\(83\) 9.28584 1.01925 0.509626 0.860396i \(-0.329784\pi\)
0.509626 + 0.860396i \(0.329784\pi\)
\(84\) 0 0
\(85\) 13.7537 1.49180
\(86\) 0 0
\(87\) 5.84093 + 2.56428i 0.626214 + 0.274920i
\(88\) 0 0
\(89\) 13.7373i 1.45615i −0.685496 0.728076i \(-0.740415\pi\)
0.685496 0.728076i \(-0.259585\pi\)
\(90\) 0 0
\(91\) 10.8803i 1.14056i
\(92\) 0 0
\(93\) −6.19451 2.71951i −0.642340 0.282000i
\(94\) 0 0
\(95\) −1.79954 −0.184629
\(96\) 0 0
\(97\) −11.2858 −1.14590 −0.572952 0.819589i \(-0.694202\pi\)
−0.572952 + 0.819589i \(0.694202\pi\)
\(98\) 0 0
\(99\) 5.41797 + 5.89298i 0.544526 + 0.592267i
\(100\) 0 0
\(101\) 9.70880i 0.966062i 0.875603 + 0.483031i \(0.160464\pi\)
−0.875603 + 0.483031i \(0.839536\pi\)
\(102\) 0 0
\(103\) 1.10098i 0.108483i −0.998528 0.0542413i \(-0.982726\pi\)
0.998528 0.0542413i \(-0.0172740\pi\)
\(104\) 0 0
\(105\) 2.92461 6.66170i 0.285413 0.650115i
\(106\) 0 0
\(107\) −18.6175 −1.79982 −0.899909 0.436077i \(-0.856368\pi\)
−0.899909 + 0.436077i \(0.856368\pi\)
\(108\) 0 0
\(109\) −12.2166 −1.17013 −0.585067 0.810985i \(-0.698932\pi\)
−0.585067 + 0.810985i \(0.698932\pi\)
\(110\) 0 0
\(111\) 0.465357 1.05999i 0.0441698 0.100610i
\(112\) 0 0
\(113\) 3.56029i 0.334924i −0.985879 0.167462i \(-0.946443\pi\)
0.985879 0.167462i \(-0.0535572\pi\)
\(114\) 0 0
\(115\) 2.04950i 0.191117i
\(116\) 0 0
\(117\) 10.7791 + 11.7241i 0.996527 + 1.08390i
\(118\) 0 0
\(119\) 13.7537 1.26080
\(120\) 0 0
\(121\) −3.87982 −0.352711
\(122\) 0 0
\(123\) 3.05589 + 1.34160i 0.275541 + 0.120968i
\(124\) 0 0
\(125\) 11.8862i 1.06313i
\(126\) 0 0
\(127\) 15.9576i 1.41601i −0.706208 0.708004i \(-0.749596\pi\)
0.706208 0.708004i \(-0.250404\pi\)
\(128\) 0 0
\(129\) −15.2858 6.71077i −1.34584 0.590851i
\(130\) 0 0
\(131\) −3.56009 −0.311047 −0.155523 0.987832i \(-0.549706\pi\)
−0.155523 + 0.987832i \(0.549706\pi\)
\(132\) 0 0
\(133\) −1.79954 −0.156040
\(134\) 0 0
\(135\) 3.44831 + 10.0758i 0.296783 + 0.867186i
\(136\) 0 0
\(137\) 7.07468i 0.604431i 0.953240 + 0.302216i \(0.0977262\pi\)
−0.953240 + 0.302216i \(0.902274\pi\)
\(138\) 0 0
\(139\) 9.51567i 0.807109i 0.914956 + 0.403554i \(0.132225\pi\)
−0.914956 + 0.403554i \(0.867775\pi\)
\(140\) 0 0
\(141\) −4.16162 + 9.47936i −0.350471 + 0.798306i
\(142\) 0 0
\(143\) 14.1657 1.18459
\(144\) 0 0
\(145\) 7.54818 0.626843
\(146\) 0 0
\(147\) −1.94921 + 4.43992i −0.160768 + 0.366198i
\(148\) 0 0
\(149\) 12.5435i 1.02760i 0.857909 + 0.513802i \(0.171763\pi\)
−0.857909 + 0.513802i \(0.828237\pi\)
\(150\) 0 0
\(151\) 8.00488i 0.651428i 0.945468 + 0.325714i \(0.105605\pi\)
−0.945468 + 0.325714i \(0.894395\pi\)
\(152\) 0 0
\(153\) −14.8205 + 13.6258i −1.19816 + 1.10158i
\(154\) 0 0
\(155\) −8.00510 −0.642985
\(156\) 0 0
\(157\) 13.9542 1.11367 0.556833 0.830624i \(-0.312016\pi\)
0.556833 + 0.830624i \(0.312016\pi\)
\(158\) 0 0
\(159\) −1.39252 0.611343i −0.110434 0.0484826i
\(160\) 0 0
\(161\) 2.04950i 0.161523i
\(162\) 0 0
\(163\) 0.438414i 0.0343392i −0.999853 0.0171696i \(-0.994534\pi\)
0.999853 0.0171696i \(-0.00546553\pi\)
\(164\) 0 0
\(165\) 8.67325 + 3.80772i 0.675212 + 0.296431i
\(166\) 0 0
\(167\) −7.28073 −0.563400 −0.281700 0.959503i \(-0.590898\pi\)
−0.281700 + 0.959503i \(0.590898\pi\)
\(168\) 0 0
\(169\) 15.1827 1.16790
\(170\) 0 0
\(171\) 1.93911 1.78281i 0.148288 0.136335i
\(172\) 0 0
\(173\) 11.9108i 0.905558i −0.891623 0.452779i \(-0.850433\pi\)
0.891623 0.452779i \(-0.149567\pi\)
\(174\) 0 0
\(175\) 1.63866i 0.123871i
\(176\) 0 0
\(177\) 10.0380 22.8646i 0.754501 1.71861i
\(178\) 0 0
\(179\) −21.5762 −1.61268 −0.806340 0.591453i \(-0.798555\pi\)
−0.806340 + 0.591453i \(0.798555\pi\)
\(180\) 0 0
\(181\) 19.2400 1.43010 0.715050 0.699073i \(-0.246404\pi\)
0.715050 + 0.699073i \(0.246404\pi\)
\(182\) 0 0
\(183\) 2.32324 5.29188i 0.171738 0.391187i
\(184\) 0 0
\(185\) 1.36982i 0.100711i
\(186\) 0 0
\(187\) 17.9068i 1.30948i
\(188\) 0 0
\(189\) 3.44831 + 10.0758i 0.250827 + 0.732906i
\(190\) 0 0
\(191\) −26.5768 −1.92303 −0.961514 0.274756i \(-0.911403\pi\)
−0.961514 + 0.274756i \(0.911403\pi\)
\(192\) 0 0
\(193\) −6.64547 −0.478351 −0.239176 0.970976i \(-0.576877\pi\)
−0.239176 + 0.970976i \(0.576877\pi\)
\(194\) 0 0
\(195\) 17.2555 + 7.57549i 1.23569 + 0.542492i
\(196\) 0 0
\(197\) 15.3484i 1.09353i 0.837287 + 0.546764i \(0.184141\pi\)
−0.837287 + 0.546764i \(0.815859\pi\)
\(198\) 0 0
\(199\) 1.21736i 0.0862967i −0.999069 0.0431483i \(-0.986261\pi\)
0.999069 0.0431483i \(-0.0137388\pi\)
\(200\) 0 0
\(201\) 19.9288 + 8.74910i 1.40567 + 0.617114i
\(202\) 0 0
\(203\) 7.54818 0.529779
\(204\) 0 0
\(205\) 3.94910 0.275817
\(206\) 0 0
\(207\) 2.03044 + 2.20846i 0.141126 + 0.153499i
\(208\) 0 0
\(209\) 2.34293i 0.162064i
\(210\) 0 0
\(211\) 10.2957i 0.708782i −0.935097 0.354391i \(-0.884688\pi\)
0.935097 0.354391i \(-0.115312\pi\)
\(212\) 0 0
\(213\) 5.87637 13.3852i 0.402642 0.917141i
\(214\) 0 0
\(215\) −19.7537 −1.34719
\(216\) 0 0
\(217\) −8.00510 −0.543422
\(218\) 0 0
\(219\) 6.15907 14.0292i 0.416191 0.948003i
\(220\) 0 0
\(221\) 35.6257i 2.39644i
\(222\) 0 0
\(223\) 8.19801i 0.548979i 0.961590 + 0.274490i \(0.0885089\pi\)
−0.961590 + 0.274490i \(0.911491\pi\)
\(224\) 0 0
\(225\) −1.62342 1.76575i −0.108228 0.117717i
\(226\) 0 0
\(227\) 16.3602 1.08587 0.542933 0.839776i \(-0.317314\pi\)
0.542933 + 0.839776i \(0.317314\pi\)
\(228\) 0 0
\(229\) 17.0286 1.12528 0.562640 0.826702i \(-0.309786\pi\)
0.562640 + 0.826702i \(0.309786\pi\)
\(230\) 0 0
\(231\) 8.67325 + 3.80772i 0.570658 + 0.250530i
\(232\) 0 0
\(233\) 1.92686i 0.126233i 0.998006 + 0.0631164i \(0.0201039\pi\)
−0.998006 + 0.0631164i \(0.979896\pi\)
\(234\) 0 0
\(235\) 12.2501i 0.799107i
\(236\) 0 0
\(237\) −5.10828 2.24263i −0.331818 0.145674i
\(238\) 0 0
\(239\) −14.3941 −0.931078 −0.465539 0.885027i \(-0.654139\pi\)
−0.465539 + 0.885027i \(0.654139\pi\)
\(240\) 0 0
\(241\) −10.0051 −0.644485 −0.322243 0.946657i \(-0.604437\pi\)
−0.322243 + 0.946657i \(0.604437\pi\)
\(242\) 0 0
\(243\) −13.6979 7.44103i −0.878718 0.477342i
\(244\) 0 0
\(245\) 5.73767i 0.366566i
\(246\) 0 0
\(247\) 4.66127i 0.296590i
\(248\) 0 0
\(249\) −6.46536 + 14.7268i −0.409725 + 0.933275i
\(250\) 0 0
\(251\) −15.9491 −1.00670 −0.503349 0.864083i \(-0.667899\pi\)
−0.503349 + 0.864083i \(0.667899\pi\)
\(252\) 0 0
\(253\) 2.66837 0.167759
\(254\) 0 0
\(255\) −9.57618 + 21.8127i −0.599684 + 1.36596i
\(256\) 0 0
\(257\) 27.3039i 1.70317i 0.524219 + 0.851584i \(0.324357\pi\)
−0.524219 + 0.851584i \(0.675643\pi\)
\(258\) 0 0
\(259\) 1.36982i 0.0851165i
\(260\) 0 0
\(261\) −8.13362 + 7.47799i −0.503458 + 0.462876i
\(262\) 0 0
\(263\) −6.61747 −0.408051 −0.204025 0.978966i \(-0.565402\pi\)
−0.204025 + 0.978966i \(0.565402\pi\)
\(264\) 0 0
\(265\) −1.79954 −0.110545
\(266\) 0 0
\(267\) 21.7866 + 9.56475i 1.33332 + 0.585353i
\(268\) 0 0
\(269\) 0.170783i 0.0104128i 0.999986 + 0.00520641i \(0.00165726\pi\)
−0.999986 + 0.00520641i \(0.998343\pi\)
\(270\) 0 0
\(271\) 28.9854i 1.76074i 0.474288 + 0.880370i \(0.342705\pi\)
−0.474288 + 0.880370i \(0.657295\pi\)
\(272\) 0 0
\(273\) 17.2555 + 7.57549i 1.04435 + 0.458490i
\(274\) 0 0
\(275\) −2.13347 −0.128653
\(276\) 0 0
\(277\) 19.2010 1.15368 0.576840 0.816857i \(-0.304286\pi\)
0.576840 + 0.816857i \(0.304286\pi\)
\(278\) 0 0
\(279\) 8.62597 7.93066i 0.516424 0.474796i
\(280\) 0 0
\(281\) 6.71077i 0.400331i 0.979762 + 0.200166i \(0.0641480\pi\)
−0.979762 + 0.200166i \(0.935852\pi\)
\(282\) 0 0
\(283\) 25.9874i 1.54479i −0.635142 0.772395i \(-0.719058\pi\)
0.635142 0.772395i \(-0.280942\pi\)
\(284\) 0 0
\(285\) 1.25295 2.85397i 0.0742183 0.169055i
\(286\) 0 0
\(287\) 3.94910 0.233108
\(288\) 0 0
\(289\) −28.0345 −1.64909
\(290\) 0 0
\(291\) 7.85788 17.8987i 0.460637 1.04924i
\(292\) 0 0
\(293\) 7.29539i 0.426201i −0.977030 0.213100i \(-0.931644\pi\)
0.977030 0.213100i \(-0.0683562\pi\)
\(294\) 0 0
\(295\) 29.5477i 1.72033i
\(296\) 0 0
\(297\) −13.1183 + 4.48955i −0.761199 + 0.260510i
\(298\) 0 0
\(299\) 5.30873 0.307012
\(300\) 0 0
\(301\) −19.7537 −1.13859
\(302\) 0 0
\(303\) −15.3976 6.75985i −0.884570 0.388343i
\(304\) 0 0
\(305\) 6.83864i 0.391580i
\(306\) 0 0
\(307\) 24.7455i 1.41230i −0.708064 0.706149i \(-0.750431\pi\)
0.708064 0.706149i \(-0.249569\pi\)
\(308\) 0 0
\(309\) 1.74609 + 0.766568i 0.0993317 + 0.0436085i
\(310\) 0 0
\(311\) 16.2394 0.920855 0.460427 0.887697i \(-0.347696\pi\)
0.460427 + 0.887697i \(0.347696\pi\)
\(312\) 0 0
\(313\) 15.2807 0.863718 0.431859 0.901941i \(-0.357858\pi\)
0.431859 + 0.901941i \(0.357858\pi\)
\(314\) 0 0
\(315\) 8.52880 + 9.27655i 0.480543 + 0.522674i
\(316\) 0 0
\(317\) 16.1507i 0.907115i 0.891227 + 0.453558i \(0.149845\pi\)
−0.891227 + 0.453558i \(0.850155\pi\)
\(318\) 0 0
\(319\) 9.82742i 0.550230i
\(320\) 0 0
\(321\) 12.9626 29.5263i 0.723502 1.64800i
\(322\) 0 0
\(323\) 5.89231 0.327857
\(324\) 0 0
\(325\) −4.24455 −0.235445
\(326\) 0 0
\(327\) 8.50590 19.3748i 0.470377 1.07143i
\(328\) 0 0
\(329\) 12.2501i 0.675369i
\(330\) 0 0
\(331\) 35.4795i 1.95013i 0.221918 + 0.975065i \(0.428768\pi\)
−0.221918 + 0.975065i \(0.571232\pi\)
\(332\) 0 0
\(333\) 1.35708 + 1.47606i 0.0743676 + 0.0808877i
\(334\) 0 0
\(335\) 25.7537 1.40708
\(336\) 0 0
\(337\) 9.54308 0.519845 0.259922 0.965630i \(-0.416303\pi\)
0.259922 + 0.965630i \(0.416303\pi\)
\(338\) 0 0
\(339\) 5.64643 + 2.47889i 0.306672 + 0.134635i
\(340\) 0 0
\(341\) 10.4223i 0.564400i
\(342\) 0 0
\(343\) 20.0842i 1.08444i
\(344\) 0 0
\(345\) 3.25040 + 1.42699i 0.174996 + 0.0768264i
\(346\) 0 0
\(347\) 24.5098 1.31575 0.657877 0.753125i \(-0.271455\pi\)
0.657877 + 0.753125i \(0.271455\pi\)
\(348\) 0 0
\(349\) −21.6639 −1.15964 −0.579820 0.814745i \(-0.696877\pi\)
−0.579820 + 0.814745i \(0.696877\pi\)
\(350\) 0 0
\(351\) −26.0989 + 8.93199i −1.39305 + 0.476755i
\(352\) 0 0
\(353\) 28.5694i 1.52059i −0.649576 0.760297i \(-0.725054\pi\)
0.649576 0.760297i \(-0.274946\pi\)
\(354\) 0 0
\(355\) 17.2976i 0.918062i
\(356\) 0 0
\(357\) −9.57618 + 21.8127i −0.506825 + 1.15445i
\(358\) 0 0
\(359\) −0.0457966 −0.00241705 −0.00120853 0.999999i \(-0.500385\pi\)
−0.00120853 + 0.999999i \(0.500385\pi\)
\(360\) 0 0
\(361\) 18.2290 0.959424
\(362\) 0 0
\(363\) 2.70136 6.15318i 0.141785 0.322958i
\(364\) 0 0
\(365\) 18.1297i 0.948954i
\(366\) 0 0
\(367\) 23.9860i 1.25206i −0.779799 0.626031i \(-0.784679\pi\)
0.779799 0.626031i \(-0.215321\pi\)
\(368\) 0 0
\(369\) −4.25539 + 3.91238i −0.221527 + 0.203670i
\(370\) 0 0
\(371\) −1.79954 −0.0934275
\(372\) 0 0
\(373\) −31.2400 −1.61755 −0.808774 0.588120i \(-0.799868\pi\)
−0.808774 + 0.588120i \(0.799868\pi\)
\(374\) 0 0
\(375\) −18.8508 8.27587i −0.973452 0.427364i
\(376\) 0 0
\(377\) 19.5517i 1.00697i
\(378\) 0 0
\(379\) 2.21282i 0.113665i −0.998384 0.0568324i \(-0.981900\pi\)
0.998384 0.0568324i \(-0.0181001\pi\)
\(380\) 0 0
\(381\) 25.3079 + 11.1106i 1.29656 + 0.569215i
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 11.2084 0.571231
\(386\) 0 0
\(387\) 21.2858 19.5701i 1.08202 0.994802i
\(388\) 0 0
\(389\) 39.0100i 1.97789i 0.148298 + 0.988943i \(0.452620\pi\)
−0.148298 + 0.988943i \(0.547380\pi\)
\(390\) 0 0
\(391\) 6.71077i 0.339378i
\(392\) 0 0
\(393\) 2.47875 5.64611i 0.125036 0.284809i
\(394\) 0 0
\(395\) −6.60138 −0.332151
\(396\) 0 0
\(397\) 30.3992 1.52569 0.762846 0.646580i \(-0.223801\pi\)
0.762846 + 0.646580i \(0.223801\pi\)
\(398\) 0 0
\(399\) 1.25295 2.85397i 0.0627259 0.142877i
\(400\) 0 0
\(401\) 7.09703i 0.354409i 0.984174 + 0.177204i \(0.0567054\pi\)
−0.984174 + 0.177204i \(0.943295\pi\)
\(402\) 0 0
\(403\) 20.7353i 1.03290i
\(404\) 0 0
\(405\) −18.3806 1.54655i −0.913338 0.0768486i
\(406\) 0 0
\(407\) 1.78345 0.0884023
\(408\) 0 0
\(409\) −5.76565 −0.285093 −0.142547 0.989788i \(-0.545529\pi\)
−0.142547 + 0.989788i \(0.545529\pi\)
\(410\) 0 0
\(411\) −11.2201 4.92582i −0.553445 0.242973i
\(412\) 0 0
\(413\) 29.5477i 1.45395i
\(414\) 0 0
\(415\) 19.0313i 0.934212i
\(416\) 0 0
\(417\) −15.0913 6.62538i −0.739026 0.324446i
\(418\) 0 0
\(419\) −1.86143 −0.0909368 −0.0454684 0.998966i \(-0.514478\pi\)
−0.0454684 + 0.998966i \(0.514478\pi\)
\(420\) 0 0
\(421\) −14.8849 −0.725447 −0.362723 0.931897i \(-0.618153\pi\)
−0.362723 + 0.931897i \(0.618153\pi\)
\(422\) 0 0
\(423\) −12.1362 13.2002i −0.590081 0.641815i
\(424\) 0 0
\(425\) 5.36554i 0.260267i
\(426\) 0 0
\(427\) 6.83864i 0.330945i
\(428\) 0 0
\(429\) −9.86298 + 22.4659i −0.476189 + 1.08467i
\(430\) 0 0
\(431\) −1.73765 −0.0836998 −0.0418499 0.999124i \(-0.513325\pi\)
−0.0418499 + 0.999124i \(0.513325\pi\)
\(432\) 0 0
\(433\) 9.59398 0.461057 0.230529 0.973066i \(-0.425954\pi\)
0.230529 + 0.973066i \(0.425954\pi\)
\(434\) 0 0
\(435\) −5.25550 + 11.9710i −0.251982 + 0.573966i
\(436\) 0 0
\(437\) 0.878038i 0.0420023i
\(438\) 0 0
\(439\) 34.4021i 1.64192i 0.570984 + 0.820961i \(0.306562\pi\)
−0.570984 + 0.820961i \(0.693438\pi\)
\(440\) 0 0
\(441\) −5.68431 6.18268i −0.270681 0.294413i
\(442\) 0 0
\(443\) −2.39411 −0.113748 −0.0568738 0.998381i \(-0.518113\pi\)
−0.0568738 + 0.998381i \(0.518113\pi\)
\(444\) 0 0
\(445\) 28.1547 1.33466
\(446\) 0 0
\(447\) −19.8933 8.73354i −0.940922 0.413082i
\(448\) 0 0
\(449\) 27.6156i 1.30326i −0.758537 0.651631i \(-0.774085\pi\)
0.758537 0.651631i \(-0.225915\pi\)
\(450\) 0 0
\(451\) 5.14157i 0.242107i
\(452\) 0 0
\(453\) −12.6953 5.57348i −0.596477 0.261865i
\(454\) 0 0
\(455\) 22.2991 1.04540
\(456\) 0 0
\(457\) −10.8747 −0.508698 −0.254349 0.967113i \(-0.581861\pi\)
−0.254349 + 0.967113i \(0.581861\pi\)
\(458\) 0 0
\(459\) −11.2909 32.9916i −0.527016 1.53992i
\(460\) 0 0
\(461\) 27.0586i 1.26024i −0.776497 0.630122i \(-0.783005\pi\)
0.776497 0.630122i \(-0.216995\pi\)
\(462\) 0 0
\(463\) 26.8431i 1.24750i 0.781622 + 0.623752i \(0.214393\pi\)
−0.781622 + 0.623752i \(0.785607\pi\)
\(464\) 0 0
\(465\) 5.57363 12.6957i 0.258471 0.588747i
\(466\) 0 0
\(467\) −20.3551 −0.941923 −0.470961 0.882154i \(-0.656093\pi\)
−0.470961 + 0.882154i \(0.656093\pi\)
\(468\) 0 0
\(469\) 25.7537 1.18920
\(470\) 0 0
\(471\) −9.71576 + 22.1306i −0.447678 + 1.01972i
\(472\) 0 0
\(473\) 25.7186i 1.18254i
\(474\) 0 0
\(475\) 0.702027i 0.0322112i
\(476\) 0 0
\(477\) 1.93911 1.78281i 0.0887859 0.0816291i
\(478\) 0 0
\(479\) −18.9217 −0.864554 −0.432277 0.901741i \(-0.642290\pi\)
−0.432277 + 0.901741i \(0.642290\pi\)
\(480\) 0 0
\(481\) 3.54818 0.161783
\(482\) 0 0
\(483\) 3.25040 + 1.42699i 0.147898 + 0.0649301i
\(484\) 0 0
\(485\) 23.1303i 1.05029i
\(486\) 0 0
\(487\) 2.14980i 0.0974167i 0.998813 + 0.0487084i \(0.0155105\pi\)
−0.998813 + 0.0487084i \(0.984490\pi\)
\(488\) 0 0
\(489\) 0.695301 + 0.305250i 0.0314426 + 0.0138039i
\(490\) 0 0
\(491\) −14.7891 −0.667425 −0.333712 0.942675i \(-0.608301\pi\)
−0.333712 + 0.942675i \(0.608301\pi\)
\(492\) 0 0
\(493\) −24.7153 −1.11312
\(494\) 0 0
\(495\) −12.0777 + 11.1041i −0.542851 + 0.499094i
\(496\) 0 0
\(497\) 17.2976i 0.775904i
\(498\) 0 0
\(499\) 15.6160i 0.699070i 0.936923 + 0.349535i \(0.113660\pi\)
−0.936923 + 0.349535i \(0.886340\pi\)
\(500\) 0 0
\(501\) 5.06929 11.5468i 0.226479 0.515875i
\(502\) 0 0
\(503\) 10.0928 0.450015 0.225007 0.974357i \(-0.427759\pi\)
0.225007 + 0.974357i \(0.427759\pi\)
\(504\) 0 0
\(505\) −19.8982 −0.885458
\(506\) 0 0
\(507\) −10.5711 + 24.0789i −0.469478 + 1.06938i
\(508\) 0 0
\(509\) 35.5981i 1.57786i −0.614483 0.788930i \(-0.710635\pi\)
0.614483 0.788930i \(-0.289365\pi\)
\(510\) 0 0
\(511\) 18.1297i 0.802013i
\(512\) 0 0
\(513\) 1.47731 + 4.31662i 0.0652247 + 0.190584i
\(514\) 0 0
\(515\) 2.25646 0.0994314
\(516\) 0 0
\(517\) −15.9491 −0.701441
\(518\) 0 0
\(519\) 18.8898 + 8.29298i 0.829170 + 0.364022i
\(520\) 0 0
\(521\) 16.6179i 0.728045i 0.931390 + 0.364022i \(0.118597\pi\)
−0.931390 + 0.364022i \(0.881403\pi\)
\(522\) 0 0
\(523\) 6.95606i 0.304167i 0.988368 + 0.152084i \(0.0485983\pi\)
−0.988368 + 0.152084i \(0.951402\pi\)
\(524\) 0 0
\(525\) −2.59883 1.14093i −0.113422 0.0497945i
\(526\) 0 0
\(527\) 26.2114 1.14179
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 29.2729 + 31.8394i 1.27034 + 1.38171i
\(532\) 0 0
\(533\) 10.2292i 0.443075i
\(534\) 0 0
\(535\) 38.1565i 1.64965i
\(536\) 0 0
\(537\) 15.0226 34.2186i 0.648275 1.47664i
\(538\) 0 0
\(539\) −7.47020 −0.321764
\(540\) 0 0
\(541\) −0.428918 −0.0184406 −0.00922031 0.999957i \(-0.502935\pi\)
−0.00922031 + 0.999957i \(0.502935\pi\)
\(542\) 0 0
\(543\) −13.3961 + 30.5136i −0.574880 + 1.30947i
\(544\) 0 0
\(545\) 25.0378i 1.07250i
\(546\) 0 0
\(547\) 34.9889i 1.49602i 0.663688 + 0.748010i \(0.268990\pi\)
−0.663688 + 0.748010i \(0.731010\pi\)
\(548\) 0 0
\(549\) 6.77505 + 7.36905i 0.289152 + 0.314503i
\(550\) 0 0
\(551\) 3.23376 0.137763
\(552\) 0 0
\(553\) −6.60138 −0.280719
\(554\) 0 0
\(555\) 2.17246 + 0.953751i 0.0922158 + 0.0404845i
\(556\) 0 0
\(557\) 11.3944i 0.482796i 0.970426 + 0.241398i \(0.0776059\pi\)
−0.970426 + 0.241398i \(0.922394\pi\)
\(558\) 0 0
\(559\) 51.1672i 2.16414i
\(560\) 0 0
\(561\) −28.3992 12.4678i −1.19902 0.526391i
\(562\) 0 0
\(563\) −27.5075 −1.15930 −0.579651 0.814865i \(-0.696811\pi\)
−0.579651 + 0.814865i \(0.696811\pi\)
\(564\) 0 0
\(565\) 7.29683 0.306980
\(566\) 0 0
\(567\) −18.3806 1.54655i −0.771911 0.0649489i
\(568\) 0 0
\(569\) 4.25146i 0.178230i −0.996021 0.0891152i \(-0.971596\pi\)
0.996021 0.0891152i \(-0.0284039\pi\)
\(570\) 0 0
\(571\) 3.66461i 0.153359i 0.997056 + 0.0766795i \(0.0244318\pi\)
−0.997056 + 0.0766795i \(0.975568\pi\)
\(572\) 0 0
\(573\) 18.5043 42.1493i 0.773030 1.76081i
\(574\) 0 0
\(575\) −0.799541 −0.0333432
\(576\) 0 0
\(577\) −23.4795 −0.977464 −0.488732 0.872434i \(-0.662540\pi\)
−0.488732 + 0.872434i \(0.662540\pi\)
\(578\) 0 0
\(579\) 4.62697 10.5393i 0.192291 0.438000i
\(580\) 0 0
\(581\) 19.0313i 0.789553i
\(582\) 0 0
\(583\) 2.34293i 0.0970342i
\(584\) 0 0
\(585\) −24.0286 + 22.0918i −0.993462 + 0.913382i
\(586\) 0 0
\(587\) −7.81342 −0.322494 −0.161247 0.986914i \(-0.551552\pi\)
−0.161247 + 0.986914i \(0.551552\pi\)
\(588\) 0 0
\(589\) −3.42951 −0.141310
\(590\) 0 0
\(591\) −24.3417 10.6865i −1.00128 0.439583i
\(592\) 0 0
\(593\) 25.3770i 1.04211i 0.853524 + 0.521054i \(0.174461\pi\)
−0.853524 + 0.521054i \(0.825539\pi\)
\(594\) 0 0
\(595\) 28.1883i 1.15561i
\(596\) 0 0
\(597\) 1.93067 + 0.847602i 0.0790172 + 0.0346900i
\(598\) 0 0
\(599\) 47.1993 1.92851 0.964256 0.264971i \(-0.0853623\pi\)
0.964256 + 0.264971i \(0.0853623\pi\)
\(600\) 0 0
\(601\) 22.6455 0.923728 0.461864 0.886951i \(-0.347181\pi\)
0.461864 + 0.886951i \(0.347181\pi\)
\(602\) 0 0
\(603\) −27.7512 + 25.5143i −1.13012 + 1.03902i
\(604\) 0 0
\(605\) 7.95169i 0.323282i
\(606\) 0 0
\(607\) 17.4609i 0.708717i −0.935110 0.354359i \(-0.884699\pi\)
0.935110 0.354359i \(-0.115301\pi\)
\(608\) 0 0
\(609\) −5.25550 + 11.9710i −0.212964 + 0.485090i
\(610\) 0 0
\(611\) −31.7308 −1.28369
\(612\) 0 0
\(613\) −34.2584 −1.38368 −0.691842 0.722049i \(-0.743201\pi\)
−0.691842 + 0.722049i \(0.743201\pi\)
\(614\) 0 0
\(615\) −2.74960 + 6.26306i −0.110875 + 0.252551i
\(616\) 0 0
\(617\) 2.27244i 0.0914851i −0.998953 0.0457426i \(-0.985435\pi\)
0.998953 0.0457426i \(-0.0145654\pi\)
\(618\) 0 0
\(619\) 29.5018i 1.18578i −0.805285 0.592889i \(-0.797987\pi\)
0.805285 0.592889i \(-0.202013\pi\)
\(620\) 0 0
\(621\) −4.91621 + 1.68251i −0.197281 + 0.0675168i
\(622\) 0 0
\(623\) 28.1547 1.12799
\(624\) 0 0
\(625\) −20.3630 −0.814521
\(626\) 0 0
\(627\) 3.71576 + 1.63129i 0.148393 + 0.0651473i
\(628\) 0 0
\(629\) 4.48526i 0.178839i
\(630\) 0 0
\(631\) 7.10972i 0.283034i −0.989936 0.141517i \(-0.954802\pi\)
0.989936 0.141517i \(-0.0451979\pi\)
\(632\) 0 0
\(633\) 16.3283 + 7.16845i 0.648993 + 0.284920i
\(634\) 0 0
\(635\) 32.7051 1.29786
\(636\) 0 0
\(637\) −14.8620 −0.588855
\(638\) 0 0
\(639\) 17.1368 + 18.6392i 0.677920 + 0.737356i
\(640\) 0 0
\(641\) 11.6151i 0.458769i −0.973336 0.229384i \(-0.926329\pi\)
0.973336 0.229384i \(-0.0736712\pi\)
\(642\) 0 0
\(643\) 29.8165i 1.17585i −0.808916 0.587925i \(-0.799945\pi\)
0.808916 0.587925i \(-0.200055\pi\)
\(644\) 0 0
\(645\) 13.7537 31.3284i 0.541553 1.23355i
\(646\) 0 0
\(647\) 2.33222 0.0916892 0.0458446 0.998949i \(-0.485402\pi\)
0.0458446 + 0.998949i \(0.485402\pi\)
\(648\) 0 0
\(649\) 38.4699 1.51007
\(650\) 0 0
\(651\) 5.57363 12.6957i 0.218448 0.497582i
\(652\) 0 0
\(653\) 41.9909i 1.64323i −0.570042 0.821616i \(-0.693073\pi\)
0.570042 0.821616i \(-0.306927\pi\)
\(654\) 0 0
\(655\) 7.29642i 0.285095i
\(656\) 0 0
\(657\) 17.9612 + 19.5359i 0.700732 + 0.762167i
\(658\) 0 0
\(659\) 7.13706 0.278020 0.139010 0.990291i \(-0.455608\pi\)
0.139010 + 0.990291i \(0.455608\pi\)
\(660\) 0 0
\(661\) −12.2726 −0.477347 −0.238673 0.971100i \(-0.576713\pi\)
−0.238673 + 0.971100i \(0.576713\pi\)
\(662\) 0 0
\(663\) −56.5004 24.8048i −2.19429 0.963337i
\(664\) 0 0
\(665\) 3.68816i 0.143021i
\(666\) 0 0
\(667\) 3.68294i 0.142604i
\(668\) 0 0
\(669\) −13.0016 5.70795i −0.502671 0.220682i
\(670\) 0 0
\(671\) 8.90363 0.343721
\(672\) 0 0
\(673\) 19.6271 0.756568 0.378284 0.925690i \(-0.376514\pi\)
0.378284 + 0.925690i \(0.376514\pi\)
\(674\) 0 0
\(675\) 3.93071 1.34524i 0.151293 0.0517781i
\(676\) 0 0
\(677\) 27.1834i 1.04474i 0.852717 + 0.522372i \(0.174953\pi\)
−0.852717 + 0.522372i \(0.825047\pi\)
\(678\) 0 0
\(679\) 23.1303i 0.887661i
\(680\) 0 0
\(681\) −11.3910 + 25.9464i −0.436503 + 0.994269i
\(682\) 0 0
\(683\) 11.4583 0.438439 0.219220 0.975676i \(-0.429649\pi\)
0.219220 + 0.975676i \(0.429649\pi\)
\(684\) 0 0
\(685\) −14.4996 −0.554001
\(686\) 0 0
\(687\) −11.8563 + 27.0064i −0.452347 + 1.03036i
\(688\) 0 0
\(689\) 4.66127i 0.177580i
\(690\) 0 0
\(691\) 52.3164i 1.99021i 0.0988314 + 0.995104i \(0.468490\pi\)
−0.0988314 + 0.995104i \(0.531510\pi\)
\(692\) 0 0
\(693\) −12.0777 + 11.1041i −0.458793 + 0.421811i
\(694\) 0 0
\(695\) −19.5024 −0.739768
\(696\) 0 0
\(697\) −12.9307 −0.489786
\(698\) 0 0
\(699\) −3.05589 1.34160i −0.115584 0.0507438i
\(700\) 0 0
\(701\) 28.5063i 1.07667i −0.842731 0.538335i \(-0.819054\pi\)
0.842731 0.538335i \(-0.180946\pi\)
\(702\) 0 0
\(703\) 0.586852i 0.0221335i
\(704\) 0 0
\(705\) −19.4280 8.52924i −0.731699 0.321230i
\(706\) 0 0
\(707\) −19.8982 −0.748349
\(708\) 0 0
\(709\) 4.18947 0.157339 0.0786694 0.996901i \(-0.474933\pi\)
0.0786694 + 0.996901i \(0.474933\pi\)
\(710\) 0 0
\(711\) 7.11338 6.53999i 0.266772 0.245269i
\(712\) 0 0
\(713\) 3.90588i 0.146276i
\(714\) 0 0
\(715\) 29.0325i 1.08576i
\(716\) 0 0
\(717\) 10.0220 22.8283i 0.374280 0.852537i
\(718\) 0 0
\(719\) 19.7537 0.736690 0.368345 0.929689i \(-0.379924\pi\)
0.368345 + 0.929689i \(0.379924\pi\)
\(720\) 0 0
\(721\) 2.25646 0.0840349
\(722\) 0 0
\(723\) 6.96615 15.8675i 0.259074 0.590120i
\(724\) 0 0
\(725\) 2.94466i 0.109362i
\(726\) 0 0
\(727\) 10.7135i 0.397342i −0.980066 0.198671i \(-0.936338\pi\)
0.980066 0.198671i \(-0.0636625\pi\)
\(728\) 0 0
\(729\) 21.3383 16.5432i 0.790308 0.612709i
\(730\) 0 0
\(731\) 64.6805 2.39230
\(732\) 0 0
\(733\) −30.3602 −1.12138 −0.560690 0.828026i \(-0.689464\pi\)
−0.560690 + 0.828026i \(0.689464\pi\)
\(734\) 0 0
\(735\) −9.09962 3.99491i −0.335645 0.147354i
\(736\) 0 0
\(737\) 33.5303i 1.23510i
\(738\) 0 0
\(739\) 12.2449i 0.450434i 0.974309 + 0.225217i \(0.0723091\pi\)
−0.974309 + 0.225217i \(0.927691\pi\)
\(740\) 0 0
\(741\) 7.39252 + 3.24546i 0.271571 + 0.119225i
\(742\) 0 0
\(743\) 21.9123 0.803885 0.401943 0.915665i \(-0.368335\pi\)
0.401943 + 0.915665i \(0.368335\pi\)
\(744\) 0 0
\(745\) −25.7079 −0.941866
\(746\) 0 0
\(747\) −18.8544 20.5074i −0.689845 0.750327i
\(748\) 0 0
\(749\) 38.1565i 1.39421i
\(750\) 0 0
\(751\) 3.15169i 0.115007i −0.998345 0.0575034i \(-0.981686\pi\)
0.998345 0.0575034i \(-0.0183140\pi\)
\(752\) 0 0
\(753\) 11.1047 25.2944i 0.404679 0.921779i
\(754\) 0 0
\(755\) −16.4060 −0.597076
\(756\) 0 0
\(757\) −47.2400 −1.71697 −0.858484 0.512840i \(-0.828593\pi\)
−0.858484 + 0.512840i \(0.828593\pi\)
\(758\) 0 0
\(759\) −1.85788 + 4.23188i −0.0674367 + 0.153608i
\(760\) 0 0
\(761\) 50.0033i 1.81262i 0.422617 + 0.906308i \(0.361111\pi\)
−0.422617 + 0.906308i \(0.638889\pi\)
\(762\) 0 0
\(763\) 25.0378i 0.906431i
\(764\) 0 0
\(765\) −27.9262 30.3746i −1.00967 1.09820i
\(766\) 0 0
\(767\) 76.5361 2.76356
\(768\) 0 0
\(769\) 15.3418 0.553241 0.276620 0.960979i \(-0.410786\pi\)
0.276620 + 0.960979i \(0.410786\pi\)
\(770\) 0 0
\(771\) −43.3024 19.0106i −1.55950 0.684649i
\(772\) 0 0
\(773\) 6.44314i 0.231744i −0.993264 0.115872i \(-0.963034\pi\)
0.993264 0.115872i \(-0.0369662\pi\)
\(774\) 0 0
\(775\) 3.12291i 0.112178i
\(776\) 0 0
\(777\) 2.17246 + 0.953751i 0.0779365 + 0.0342156i
\(778\) 0 0
\(779\) 1.69186 0.0606170
\(780\) 0 0
\(781\) 22.5208 0.805857
\(782\) 0 0
\(783\) −6.19657 18.1061i −0.221447 0.647059i
\(784\) 0 0
\(785\) 28.5992i 1.02075i
\(786\) 0 0
\(787\) 45.4072i 1.61859i 0.587400 + 0.809297i \(0.300151\pi\)
−0.587400 + 0.809297i \(0.699849\pi\)
\(788\) 0 0
\(789\) 4.60748 10.4949i 0.164031 0.373630i
\(790\) 0 0
\(791\) 7.29683 0.259445
\(792\) 0 0
\(793\) 17.7138 0.629037
\(794\) 0 0
\(795\) 1.25295 2.85397i 0.0444375 0.101220i
\(796\) 0 0
\(797\) 48.8030i 1.72869i −0.502898 0.864346i \(-0.667733\pi\)
0.502898 0.864346i \(-0.332267\pi\)
\(798\) 0 0
\(799\) 40.1110i 1.41902i
\(800\) 0 0
\(801\) −30.3383 + 27.8929i −1.07195 + 0.985545i
\(802\) 0 0
\(803\) 23.6042 0.832974
\(804\) 0 0
\(805\) 4.20046 0.148047
\(806\) 0 0
\(807\) −0.270852 0.118909i −0.00953445 0.00418581i
\(808\) 0 0
\(809\) 47.2338i 1.66065i 0.557278 + 0.830326i \(0.311846\pi\)
−0.557278 + 0.830326i \(0.688154\pi\)
\(810\) 0 0
\(811\) 3.45999i 0.121497i −0.998153 0.0607484i \(-0.980651\pi\)
0.998153 0.0607484i \(-0.0193487\pi\)
\(812\) 0 0
\(813\) −45.9693 20.1814i −1.61221 0.707792i
\(814\) 0 0
\(815\) 0.898531 0.0314742
\(816\) 0 0
\(817\) −8.46281 −0.296076
\(818\) 0 0
\(819\) −24.0286 + 22.0918i −0.839628 + 0.771949i
\(820\) 0 0
\(821\) 27.2740i 0.951871i 0.879480 + 0.475935i \(0.157890\pi\)
−0.879480 + 0.475935i \(0.842110\pi\)
\(822\) 0 0
\(823\) 23.6650i 0.824911i 0.910978 + 0.412456i \(0.135329\pi\)
−0.910978 + 0.412456i \(0.864671\pi\)
\(824\) 0 0
\(825\) 1.48545 3.38356i 0.0517167 0.117801i
\(826\) 0 0
\(827\) −31.8066 −1.10602 −0.553012 0.833173i \(-0.686522\pi\)
−0.553012 + 0.833173i \(0.686522\pi\)
\(828\) 0 0
\(829\) 33.3070 1.15680 0.578400 0.815753i \(-0.303677\pi\)
0.578400 + 0.815753i \(0.303677\pi\)
\(830\) 0 0
\(831\) −13.3689 + 30.4518i −0.463763 + 1.05636i
\(832\) 0 0
\(833\) 18.7871i 0.650934i
\(834\) 0 0
\(835\) 14.9219i 0.516393i
\(836\) 0 0
\(837\) 6.57167 + 19.2021i 0.227150 + 0.663723i
\(838\) 0 0
\(839\) 27.0337 0.933307 0.466653 0.884440i \(-0.345460\pi\)
0.466653 + 0.884440i \(0.345460\pi\)
\(840\) 0 0
\(841\) 15.4360 0.532275
\(842\) 0 0
\(843\) −10.6429 4.67244i −0.366562 0.160928i
\(844\) 0 0
\(845\) 31.1169i 1.07045i
\(846\) 0 0
\(847\) 7.95169i 0.273223i
\(848\) 0 0
\(849\) 41.2146 + 18.0940i 1.41448 + 0.620984i
\(850\) 0 0
\(851\) 0.668367 0.0229113
\(852\) 0 0
\(853\) 33.7698 1.15626 0.578129 0.815946i \(-0.303783\pi\)
0.578129 + 0.815946i \(0.303783\pi\)
\(854\) 0 0
\(855\) 3.65387 + 3.97422i 0.124960 + 0.135915i
\(856\) 0 0
\(857\) 27.6454i 0.944350i −0.881505 0.472175i \(-0.843469\pi\)
0.881505 0.472175i \(-0.156531\pi\)
\(858\) 0 0
\(859\) 1.46667i 0.0500421i 0.999687 + 0.0250210i \(0.00796528\pi\)
−0.999687 + 0.0250210i \(0.992035\pi\)
\(860\) 0 0
\(861\) −2.74960 + 6.26306i −0.0937062 + 0.213444i
\(862\) 0 0
\(863\) −50.2406 −1.71021 −0.855105 0.518454i \(-0.826508\pi\)
−0.855105 + 0.518454i \(0.826508\pi\)
\(864\) 0 0
\(865\) 24.4111 0.830003
\(866\) 0 0
\(867\) 19.5193 44.4611i 0.662910 1.50998i
\(868\) 0 0
\(869\) 8.59472i 0.291556i
\(870\) 0 0
\(871\) 66.7088i 2.26034i
\(872\) 0 0
\(873\) 22.9153 + 24.9243i 0.775564 + 0.843561i
\(874\) 0 0
\(875\) −24.3607 −0.823543
\(876\) 0 0
\(877\) 25.3367 0.855561 0.427780 0.903883i \(-0.359296\pi\)
0.427780 + 0.903883i \(0.359296\pi\)
\(878\) 0 0
\(879\) 11.5701 + 5.07949i 0.390249 + 0.171327i
\(880\) 0 0
\(881\) 3.68919i 0.124292i −0.998067 0.0621460i \(-0.980206\pi\)
0.998067 0.0621460i \(-0.0197945\pi\)
\(882\) 0 0
\(883\) 57.4928i 1.93479i −0.253279 0.967393i \(-0.581509\pi\)
0.253279 0.967393i \(-0.418491\pi\)
\(884\) 0 0
\(885\) 46.8610 + 20.5729i 1.57522 + 0.691550i
\(886\) 0 0
\(887\) 46.3542 1.55642 0.778211 0.628003i \(-0.216127\pi\)
0.778211 + 0.628003i \(0.216127\pi\)
\(888\) 0 0
\(889\) 32.7051 1.09690
\(890\) 0 0
\(891\) 2.01354 23.9307i 0.0674562 0.801710i
\(892\) 0 0
\(893\) 5.24812i 0.175622i
\(894\) 0 0
\(895\) 44.2204i 1.47813i
\(896\) 0 0
\(897\) −3.69626 + 8.41936i −0.123415 + 0.281114i
\(898\) 0 0
\(899\) 14.3851 0.479770
\(900\) 0 0
\(901\) 5.89231 0.196301
\(902\) 0 0
\(903\) 13.7537 31.3284i 0.457696 1.04254i
\(904\) 0 0
\(905\) 39.4325i 1.31078i
\(906\) 0 0
\(907\) 21.0235i 0.698072i −0.937109 0.349036i \(-0.886509\pi\)
0.937109 0.349036i \(-0.113491\pi\)
\(908\) 0 0
\(909\) 21.4415 19.7132i 0.711170 0.653845i
\(910\) 0 0
\(911\) 8.85784 0.293473 0.146737 0.989176i \(-0.453123\pi\)
0.146737 + 0.989176i \(0.453123\pi\)
\(912\) 0 0
\(913\) −24.7780 −0.820033
\(914\) 0 0
\(915\) 10.8457 + 4.76148i 0.358548 + 0.157410i
\(916\) 0 0
\(917\) 7.29642i 0.240949i
\(918\) 0 0
\(919\) 27.8152i 0.917538i −0.888556 0.458769i \(-0.848291\pi\)
0.888556 0.458769i \(-0.151709\pi\)
\(920\) 0 0
\(921\) 39.2449 + 17.2293i 1.29316 + 0.567724i
\(922\) 0 0
\(923\) 44.8052 1.47478
\(924\) 0 0
\(925\) −0.534387 −0.0175705
\(926\) 0 0
\(927\) −2.43147 + 2.23548i −0.0798599 + 0.0734227i
\(928\) 0 0
\(929\) 34.5654i 1.13405i 0.823699 + 0.567027i \(0.191907\pi\)
−0.823699 + 0.567027i \(0.808093\pi\)
\(930\) 0 0
\(931\) 2.45810i 0.0805611i
\(932\) 0 0
\(933\) −11.3069 + 25.7549i −0.370171 + 0.843177i
\(934\) 0 0
\(935\) −36.7000 −1.20022
\(936\) 0 0
\(937\) 27.5392 0.899665 0.449833 0.893113i \(-0.351484\pi\)
0.449833 + 0.893113i \(0.351484\pi\)
\(938\) 0 0
\(939\) −10.6394 + 24.2344i −0.347203 + 0.790860i
\(940\) 0 0
\(941\) 40.6624i 1.32556i 0.748816 + 0.662778i \(0.230623\pi\)
−0.748816 + 0.662778i \(0.769377\pi\)
\(942\) 0 0
\(943\) 1.92686i 0.0627472i
\(944\) 0 0
\(945\) −20.6504 + 7.06731i −0.671756 + 0.229900i
\(946\) 0 0
\(947\) −13.0057 −0.422628 −0.211314 0.977418i \(-0.567774\pi\)
−0.211314 + 0.977418i \(0.567774\pi\)
\(948\) 0 0
\(949\) 46.9607 1.52441
\(950\) 0 0
\(951\) −25.6142 11.2451i −0.830596 0.364648i
\(952\) 0 0
\(953\) 32.7687i 1.06148i −0.847534 0.530740i \(-0.821914\pi\)
0.847534 0.530740i \(-0.178086\pi\)
\(954\) 0 0
\(955\) 54.4692i 1.76258i
\(956\) 0 0
\(957\) −15.5858 6.84244i −0.503816 0.221185i
\(958\) 0 0
\(959\) −14.4996 −0.468216
\(960\) 0 0
\(961\) 15.7441 0.507875
\(962\) 0 0
\(963\) 37.8017 + 41.1159i 1.21814 + 1.32494i
\(964\) 0 0
\(965\) 13.6199i 0.438440i
\(966\) 0 0
\(967\) 54.8479i 1.76379i 0.471445 + 0.881895i \(0.343733\pi\)
−0.471445 + 0.881895i \(0.656267\pi\)
\(968\) 0 0
\(969\) −4.10258 + 9.34489i −0.131794 + 0.300201i
\(970\) 0 0
\(971\) −9.80543 −0.314671 −0.157336 0.987545i \(-0.550290\pi\)
−0.157336 + 0.987545i \(0.550290\pi\)
\(972\) 0 0
\(973\) −19.5024 −0.625218
\(974\) 0 0
\(975\) 2.95531 6.73162i 0.0946457 0.215585i
\(976\) 0 0
\(977\) 6.64150i 0.212480i 0.994340 + 0.106240i \(0.0338812\pi\)
−0.994340 + 0.106240i \(0.966119\pi\)
\(978\) 0 0
\(979\) 36.6562i 1.17154i
\(980\) 0 0
\(981\) 24.8050 + 26.9798i 0.791963 + 0.861398i
\(982\) 0 0
\(983\) 6.70906 0.213986 0.106993 0.994260i \(-0.465878\pi\)
0.106993 + 0.994260i \(0.465878\pi\)
\(984\) 0 0
\(985\) −31.4566 −1.00229
\(986\) 0 0
\(987\) −19.4280 8.52924i −0.618399 0.271489i
\(988\) 0 0
\(989\) 9.63831i 0.306481i
\(990\) 0 0
\(991\) 0.245286i 0.00779177i −0.999992 0.00389588i \(-0.998760\pi\)
0.999992 0.00389588i \(-0.00124010\pi\)
\(992\) 0 0
\(993\) −56.2686 24.7030i −1.78563 0.783925i
\(994\) 0 0
\(995\) 2.49499 0.0790965
\(996\) 0 0
\(997\) 29.6977 0.940537 0.470269 0.882523i \(-0.344157\pi\)
0.470269 + 0.882523i \(0.344157\pi\)
\(998\) 0 0
\(999\) −3.28584 + 1.12453i −0.103959 + 0.0355787i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1104.2.e.f.47.4 yes 8
3.2 odd 2 1104.2.e.g.47.6 yes 8
4.3 odd 2 1104.2.e.g.47.5 yes 8
12.11 even 2 inner 1104.2.e.f.47.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1104.2.e.f.47.3 8 12.11 even 2 inner
1104.2.e.f.47.4 yes 8 1.1 even 1 trivial
1104.2.e.g.47.5 yes 8 4.3 odd 2
1104.2.e.g.47.6 yes 8 3.2 odd 2